Unit cell dimensions of crystal structure

[MightyQuinn11]

Hello all,

I already have the answer to my question, but what I would like to know is why. My question is: why in the question below is the plane 111 used to find ao and not another plane?

An ordered compound of NiAl has a cubic structure with one formula unit/cell. One Al atom at the origin and one Ni at 1/2 1/2 1/2. The nine lines of lowest bragg angle on a diffraction pattern have d-spacing of: 4.05, 2.86, 2.34, 2.03, 1.81, 1.65, 1.43, 1.35, 1.28.

QUESTION: Evaluate the unit cell dimensions of NiAl.

the formula to use is: dhkl=ao/sqrt(h^2+k^2+l^2)

The answer is: ao= sqrt(3)*2.34= 4.05A

Thank you for any clarification on this topic.

 

[OutdoorsDude]

The formula you have comes form the unit cell being cubic. Using that formula you can find the lattice parameter using any of the planes (100, 110, 111, 200, 210, etc). They will all give you the same lattice parameter for the unit cell.

The easiest way would be to use the (100) plane seeing as a0 = sqrt(1)*4.05 = 4.05 A.

 

[MightyQuinn11]

Thank you, this is very useful. Now, if it was not cubic would I have to select another plane? Maybe a plane that contains a peak?

 

[Gokul43201]

If it were not cubic, would you still have the same formula for the d-spacing? If not, what part of it would change?

 

[MightyQuinn11]

It comes from the same formula, for orthogonal axes the formula is 1/d^2=h^2/a^2+k^2/b^2+l^2/c^2. Since cubic has a=b=c, b and c are not needed in the formula reducing it to the formula below in my previous post. Now, if it is not orthogonal, the formula changes. I cannot find all of the formulas for the other structures.

I am now thinking the reason they omit certain planes is because they would give the same result? For example, in BCC the planes 100, 010, 001 and 110 give the same d-spacing result? If not, why do they omit the first 3 planes only listing 110?

Correct me if I am wrong and help with any underlying physics that i am not clear enough on. Thank you for any help.

 

[Gokul43201]

It comes from the same formula, for orthogonal axes the formula is 1/d^2=h^2/a^2+k^2/b^2+l^2/c^2. Since cubic has a=b=c, b and c are not needed in the formula reducing it to the formula below in my previous post. Now, if it is not orthogonal, the formula changes.

Correct.

I am now thinking the reason they omit certain planes is because they would give the same result? For example, in BCC the planes 100, 010, 001 and 110 give the same d-spacing result? If not, why do they omit the first 3 planes only listing 110?

You can check very easily that in a bcc lattice (or most any lattice, for that matter), the (100) planes have a different d-spacing than the (110) planes. There is a different reason that the (100) family of planes do not contribute Bragg peaks in the bcc lattice. Do you know what the structure factor is, and have you calculated it for different lattices? If not, you should look that up.